An edge coloring of a graph is a local r -coloring if the edges incident to any vertex are colored with at most r distinct colors. In this paper, generalizing our earlier work, we study the following problem. Given a family of graphs $$\mathcal {F} $$ F (for example matchings, paths, cycles, powers of cycles and paths, connected subgraphs) and fixed positive integers s , r , at least how many vertices can be covered by the vertices of no more than s monochromatic members of $$\mathcal {F} $$ F in every local r -coloring of $$K_n$$ K n . Several problems and results are presented. In particular, we prove the following two results. First, if n is sufficiently large then in any local r -coloring of the edges of $$K_n$$ K n , the vertex set can be partitioned by the vertices of at most r monochromatic trees, which is sharp for local r -colorings (unlike for ordinary r -colorings according to the Ryser conjecture). Second, we show that we can partition the vertex set with at most O ( r log r ) monochromatic cycles in every local r -coloring of $$K_n$$ K n . This answers a question of Conlon and Stein and slightly generalizes one of my favorite joint results with Endre (and with Gyárfás and Ruszinkó).

中文翻译：

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局部边缘着色中的单色分区

如果入射到任何顶点的边最多用 r 种不同颜色着色，则图的边着色是局部 r 着色。在本文中，概括我们早期的工作，我们研究以下问题。给定一组图 $$\mathcal {F} $$ F（例如匹配、路径、循环、循环和路径的幂、连通子图）和固定正整数 s 、 r ，至少有多少顶点可以被覆盖$$\mathcal {F} $$ F 在 $$K_n$$ K n 的每个局部 r 着色中不超过 s 个单色成员的顶点。提出了几个问题和结果。特别地，我们证明了以下两个结果。首先，如果 n 足够大，那么在 $$K_n$$ K n 边缘的任何局部 r 着色中，顶点集可以由至多 r 个单色树的顶点划分，这对于局部 r 着色很明显（与根据 Ryser 猜想的普通 r 着色不同）。其次，我们证明我们可以在 $$K_n$$K n 的每个局部 r 着色中用最多 O ( r log r ) 个单色循环来划分顶点集。这回答了 Conlon 和 Stein 的问题，并稍微概括了我与 Endre（以及 Gyárfás 和 Ruszinkó）最喜欢的联合结果之一。